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REGULARIZING PRIORS FOR LINEAR INVERSE PROBLEMS

Published online by Cambridge University Press:  06 November 2014

Jean-Pierre Florens  and
Anna Simoni
Jean-Pierre Florens
Affiliation:
Toulouse School of Economics
Anna Simoni*
Affiliation:
CNRS – GRECSTA and CREST
*
*Address correspondence to Anna Simoni, CREST, 15 Boulevard Gabriel Péri, 92240 Malakoff, France; e-mail: [email protected].
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Abstract

This paper proposes a new Bayesian approach for estimating, nonparametrically, functional parameters in econometric models that are characterized as the solution of a linear inverse problem. By using a Gaussian process prior we propose the posterior mean as an estimator and prove frequentist consistency of the posterior distribution. The latter provides the frequentist validation of our Bayesian procedure. We show that the minimax rate of contraction of the posterior distribution can be obtained provided that either the regularity of the prior matches the regularity of the true parameter or the prior is scaled at an appropriate rate. The scaling parameter of the prior distribution plays the role of a regularization parameter. We propose a new data-driven method for optimally selecting in practice this regularization parameter. We also provide sufficient conditions such that the posterior mean, in a conjugate-Gaussian setting, is equal to a Tikhonov-type estimator in a frequentist setting. Under these conditions our data-driven method is valid for selecting the regularization parameter of the Tikhonov estimator as well. Finally, we apply our general methodology to two leading examples in econometrics: instrumental regression and functional regression estimation.

Type
ARTICLES
Information
Econometric Theory , Volume 32 , Issue 1 , February 2016 , pp. 71 - 121
Copyright
Copyright © Cambridge University Press 2014 

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References

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